document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191339>Question 1191339</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Assume that a procedure yields a binomial distribution with a trial repeated  <BR>\n" );
document.write( "n=18 times. Use either the binomial probability formula (or technology) to find the probability of  k=4 successes given the probability  p=0.41 of success on a single trial.\r<BR>\n" );
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document.write( "(Report answer accurate to 4 decimal places.)\r<BR>\n" );
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document.write( "2. Assume that a procedure yields a binomial distribution with a trial repeated  <BR>\n" );
document.write( "n=11times. Use the binomial probability formula to find the probability of  k=7 successes given the probability  p= 2/3   of success on a single trial.\r<BR>\n" );
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document.write( "(Report answer accurate to 4 decimal places.)\r<BR>\n" );
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document.write( "3. Suppose that a box contains 7 cameras and that 4 of them are defective. A sample of 2 cameras is selected at random with replacement. Define the random variable  X as the number of defective cameras in the sample.\r<BR>\n" );
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document.write( "a. Write the binomial probability distribution for  X. Round to two decimal places.\r<BR>\n" );
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document.write( "b. What is the expected value of  X? Round to two decimal places.\r<BR>\n" );
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document.write( "4. An insurance company prices its Tornado Insurance using the following assumptions:\r<BR>\n" );
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document.write( "In any calendar year, there can be at most one tornado.<BR>\n" );
document.write( "In any calendar year, the probability of a tornado is 0.11.<BR>\n" );
document.write( "The number of tornadoes in any calendar year is independent of the number of tornados in any other calendar year.<BR>\n" );
document.write( "Using the insurance company's assumptions, calculate the probability that there are fewer than 4 tornadoes in a 23-year period. Round your answer to four decimal places.\r<BR>\n" );
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document.write( "5. A salesperson makes 7 sales per day on average. Use the formula for Poisson probabilities\r<BR>\n" );
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document.write( "to find the probability that the salesperson makes exactly 6 sales on a given day.\r<BR>\n" );
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document.write( "(a) First fill in the details of the binomial probability formula:\r<BR>\n" );
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document.write( "P<BR>\n" );
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document.write( "6<BR>\n" );
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document.write( "(b) The probability, rounded to 4 decimal places, is \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849178 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849178\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are the solutions to your probability problems:\r<BR>\n" );
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document.write( "**1. Binomial Probability (n=18, k=4, p=0.41):**\r<BR>\n" );
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document.write( "The binomial probability formula is:\r<BR>\n" );
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document.write( "P(k) = (nCk) * p^k * (1-p)^(n-k)\r<BR>\n" );
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document.write( "Where nCk is the binomial coefficient, calculated as n! / (k! * (n-k)!).\r<BR>\n" );
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document.write( "P(4) = (18C4) * (0.41)^4 * (1 - 0.41)^(18-4)<BR>\n" );
document.write( "P(4) = 3060 * 0.028257761 * 0.006553568<BR>\n" );
document.write( "P(4) ≈ 0.0567\r<BR>\n" );
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document.write( "**2. Binomial Probability (n=11, k=7, p=2/3):**\r<BR>\n" );
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document.write( "P(7) = (11C7) * (2/3)^7 * (1 - 2/3)^(11-7)<BR>\n" );
document.write( "P(7) = 330 * (128/2187) * (1/81)<BR>\n" );
document.write( "P(7) ≈ 0.2376\r<BR>\n" );
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document.write( "**3. Binomial Probability Distribution (n=2, 7 cameras, 4 defective):**\r<BR>\n" );
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document.write( "Since the cameras are selected *with* replacement, we use the binomial distribution.  The probability of selecting a defective camera is 4/7, and the probability of selecting a non-defective camera is 3/7.\r<BR>\n" );
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document.write( "a. **Binomial Probability Distribution for X:**\r<BR>\n" );
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document.write( "* P(X=0) = (2C0) * (4/7)^0 * (3/7)^2 = 1 * 1 * 9/49 ≈ 0.18<BR>\n" );
document.write( "* P(X=1) = (2C1) * (4/7)^1 * (3/7)^1 = 2 * 4/7 * 3/7 = 24/49 ≈ 0.49<BR>\n" );
document.write( "* P(X=2) = (2C2) * (4/7)^2 * (3/7)^0 = 1 * 16/49 * 1 = 16/49 ≈ 0.33\r<BR>\n" );
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document.write( "b. **Expected Value of X:**\r<BR>\n" );
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document.write( "E(X) = n * p = 2 * (4/7) = 8/7 ≈ 1.14\r<BR>\n" );
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document.write( "**4. Probability of Fewer Than 4 Tornadoes in 23 Years:**\r<BR>\n" );
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document.write( "Since tornadoes are independent and the probability of a tornado in a year is constant, we can model this using the binomial distribution with n = 23 and p = 0.11. We want P(X < 4), which is P(X=0) + P(X=1) + P(X=2) + P(X=3).\r<BR>\n" );
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document.write( "Using a binomial calculator or software is highly recommended for this.  Calculating each term and summing them, we get:\r<BR>\n" );
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document.write( "P(X < 4) ≈ 0.0785 + 0.2022 + 0.2606 + 0.2103 ≈ 0.7516\r<BR>\n" );
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document.write( "**5. Poisson Probability (λ=7, k=6):**\r<BR>\n" );
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document.write( "a. **Poisson Probability Formula Details:**\r<BR>\n" );
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document.write( "P(6) = (7^6 * e^-7) / 6!\r<BR>\n" );
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document.write( "b. **Probability:**\r<BR>\n" );
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document.write( "P(6) = (117649 * 0.00091188) / 720<BR>\n" );
document.write( "P(6) ≈ 0.1490<BR>\n" );
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